
where \(\mathcal {F}_{\ell , \pm },\mathcal {G}_{\ell , \pm },\mathcal {H}_{\ell , \pm }\)  are polynomials
and \(\delta _+=1, \delta _-=0\) . This yields
the local smooth solution
\(f_{\ell ,\pm }(0,t_{\underline{r}}),g_{\ell ,\pm }(0,t_{\underline{r}}),f_{\ell ,\pm }(0,t_{\underline{r}})\) 
on some open interval \(\mathcal {I}_2\)  with \(0\in \mathcal {I}_2.\)  Noticing that
the Forkas-Lenells hierarchy ([1]}, (2.37))
is equivalent to the following
first-order system (taking \(x=0\)  here)
\(q_{t_{\underline{r}}}(x,t_{\underline{r}})=&f_{n_-,-}(x,t_r)-f_{n_+-1,+}(x,t_{\underline{r}})+C_1,\\r_{t_{\underline{r}}}(x,t_{\underline{r}})=&h_{n_+-1,+}(x,t_r)-f_{n_-,-}(x,t_{\underline{r}})+C_2, \quad C_1, C_2\in \mathbb {C},\) 
